Nonlinear Forced Response of Infinitely Long Circular Cylindrical Shells

A combination of the Galerkin procedure and the method of multiple scales is used to analyze the nonlinear forced response of infinitely long circular cylindrical shells (or circular rings) in the presence of internal (autoparametric) resonances. If ω_f and a_f denote the frequency and amplitude of a flexural mode and ω_b and a_b denote the frequency and amplitude of the breathing mode, the steady-state response is found to exhibit a saturation phenomenon when ω_b = 2ω_f if the shell is excited by a harmonic load having a frequency Ω near ω_b. As the amplitude f of the excitation increases from zero, a_b increases linearly with f until a threshold value f_c of f is reached. Beyond f_c, a_b remains constant and the extra energy spills over into the flexural resonant mode whose amplitude grows nonlinearly. Results of numerical investigations, guided by the perturbation analysis, show that the long-time response exhibits a Hopf bifurcation, yielding amplitude and phase-modulated motions. The amplitudes and phases experience a cascade of period-doubling bifurcations ending up with chaos. The bifurcation values are finely tuned.